Methods and apparatus for generating exponential and power functions



ay 16, 1967 H. M. MARTINEZ 3,320,411

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METHODS AND APPARATUS FOR GENERATING EXPONENTIAL AND POWER FUNCTIONSOriginal Filed May 11, 1959 10 Sheets-Sheet 8 y X R x =--X x 2 Q 524xfio or x o 52 INVEN TOR.

HUGO M MARTINEZ A T TORNEV May 16. 1967 H. M. MART!NEZ 3,320,411

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I 1(a --wv '-Tf/ HUGO M. MARTINEZ AT TOPNEV United States Patent3,320,411 METHODS AND APPARATUS FOR GENERATING EXPQNENTIAL AND POWERFUNCTIONS Hugo M. Martinez, Chicago, 111., assignor to Yuba ConsolidatedIndustries, Inc., San Francisco, Calif., a corporation of DelawareOriginal application May 11, 1959, Ser. No. 812,566, now Patent No.3,259,736, dated July 5, 1966. Divided and this application Dec. 10,1962, Ser. No. 243,689 7 Claims. (Cl. 235-197) The invention describedherein may be manufactured and used by or for the Government of theUnited States of America for governmental purposes without the paymentof any royalties thereon or therefor.

This application is a division of my copending application Ser. No.812,566, filed May 11, 1959, now Patent No. 3,259,736.

This invention relates to methods and apparatus for producing physicalquantities representative of mathe matical functions. More particularlythis invention relates to methods and apparatus for generatingexponential and power functions.

Computing devices and other equipment often require the generation ofphysical quantities such as voltages, currents, displacements, or thelike, representative of various mathematical functions. Arrangements foraccomplishing these purposes are commonly known as function generators.Prior art function generators often have been complicated and difficultto construct. An object of the present invention is to providerelatively simple, reliable, and accurate apparatus and methods forgenerating functions, particularly exponential and power functions.

Other objects and many of the attendant advantages of this inventionwill be readily appreciated as the same becomes better understood byreference to the following detailed description when considered inconnection with the accompanying drawings wherein:

FIG. 1 is a graph illustrating a periodic time representation of -alinear function having a duty cycle less than 100%;

FIG. 2 is a block diagram showing one apparatus of this invention;

FIG. 3 is a graph illustrating a periodic time representation of alinear function having a 100% duty cycle;

FIG. 4 is a graph illustrating a periodic time representation of themonotonic segment of the sine function having a 100% duty cycle;

FIG. 5 is a block diagram of an apparatus for generating the arc sinefunction;

FIG. 6 is a schematic diagram of an apparatus for generating the sineand cosine functions using the apparatus of FIG. 5 in the feedback of anamplifier;

FIG. 7 is a block diagram of an apparatus for generating the sine andcosine functions using the basic method of the invention;

FIG. 8 is a graph illustrating a periodic time representation of the arcsine function produced from a sine wave;

FIG. 9 is a graph illustrating the static function set-up used to modifya sine wave to produce the graph of FIG. 8;

FIG. 10 is a schematic diagram of an apparatus, using an arc sinegenerator in the feedback of an amplifier, for producing the sinefunction and cosine function for a range of angles extending over 371'radians;

FIG. 11 is a graph illustrating the operation of the means in FIG. 10for extending the usable angular range;

FIG. 12 is a schematic diagram of an apparatus using the basic method ofthe invention for generating the sine and cosine functions with meansextending the angular range over 31r radians;

FIG. 13 is a schematic diagram of an apparatus used r r 3,320,411Patented May 16, 1967 for extending without limit the angular range ofsine and cosine generators;

FIG. 14 is a schematic diagram of an apparatus for accomplishing polarto rectangular transformations using arc sine generators in the feedbackof amplifiers;

FIG. 15 is a schematic diagram of an apparatus for accomplishing polarto rectangular transformations by direct application of the basic methodof the invention;

FIG. 16 is a schematic diagram of a four-quadrant multiplier;

FIG. 17 is a schematic diagram of an apparatus for generating a periodictime representation of a positive exponential;

FIG. 18 is a schematic diagram of an apparatus using the apparatus ofFIG. 17 in the basic method of the invention for generating alogarithmic function;

FIG. 19 is a schematic diagram of an apparatus using the apparatus ofFIG. 18 in the feedback of an amplifier for generating an exponentialfunction;

FIG. 20 is a schematic diagram of an apparatus for generatingessentially a periodic time representation of a negative exponential;

FIG. 21 is a schematic diagram of an apparatus using the apparatus ofFIG. 20 and the basic method of the invention to generate the logarithmof reciprocals;

FIG. 22 is a schematic diagram of a circuit using the apparatus of FIG.21 in the feedback of an amplifier for producing negative exponentials;

FIG. 23 is a schematic diagram of a circuit using the apparatus of FIG.18 for producing positive constant powers of a variable;

FIG. 24 is a schematic diagram of a circuit using the apparatus of FIG.18 and of FIG. 21 for generating negative constant powers of a variable;

FIG. 25 is a schematic diagram of a circuit using the apparatus of FIG.18 for generating variable powers of a variable;

FIG. 26 is a schematic diagram of an apparatus for producing periodictime representations of a linear function, a quadratic function, a cubicfunction, etc.;

FIGS. 27(a) and (b) show schematically the diagram of circuits using theapparatus of FIG. 26 for generating square roots and cube roots,respectively;

FIGS. 28 (a) and (b) show schematically the diagram of circuits usingthe apparatus of FIGS. 27(a) and (b), respectively, for the generationof squares and cubes, respectively;

FIG. 29 is a schematic diagram of an apparatus using trigonometricrelations to generate constant powers without the use of logarithms.

The methods and apparatus of the invention are based on what arebelieved to be certain novel mathematical relations. For an adequateunderstanding of the invention, an exposition of these relations isfirst set forth herewith.

A function is a quantity which takes on a definite value, or values,when special values are assigned to certain quantities, called thearguments or independent variables of the function. Examples offunctions of one variable, x, are the following: 2x; lx sin x; e; log x.These are also called functional expressions. One quantity is said to bea function of another if to each value of the second (the independentvariable) there corresponds a value of the first (the dependentvariable). The range of the independent variable is either explicitlystated, or understood from the context. The foregoing examples offunctional expressions are specific functions of x. The symbols used fora general function of x are f(x), g(x), F(x), (x), etc. Such symbols areused when making statements that are true for several differentfunctions, in other words, statements that are not concerned with aspecific form of function.

Frequently a single symbol, constituting the independent variable, isused to represent a function and is then defined as equal to theparticular, specific functional expression in the dependent variable orto the general function. Thus, for example, where the symbol y is usedto represent a function it may, using the previous expressions asexamples, be defined specifically as y=2x; y=(1x y=sin x; etc., or itmay be defined in the case of a general function as y=f(x); y= em Aninverse function or the inverse of a function is the function obtainedby expressing the independent variable explicitly in terms of thedependent variable and considering the dependent variable as anindependent variable. If y=f(x) results in x=g(y), the latter is theinverse of the former (and vice versa). Thus where a function y isdefined as y=2x, the inverse function is x= /zy. In the case of thegeneral function where y=f(x), the inverse function is written x=f* (y).

It must be remembered that a function is always regarded as beingconfined within limits constituting the range of interest. That is,there are limiting values to the function which depend on eitherexplicitly expressed limits of the dependent variable or, impliedly,those limits of the dependent variable for which the function isdefined.

A function generator is an apparatus which, assuming the functionalrelation between two variables, for example, to be expressed by y=f(x),will, when supplied with any particular value of x, say x within thelimits of the function produce the corresponding value of y, say y Thisprocess of producing from a given value of x the corresponding value ofy is called generating a function.

Denoting in general a functional relation between two variables by y=(x) and the inverse function by x=f (y), the method of the presentinvention achieves the automatic physical realization of the relationy=f(x) by the use of the relation x=f- (y). This means that given aspecific value of x in some physical form such as a voltage, current, orthe like, then the corresponding value of y will be generated in thesame or analogous physical form, using the relation x=f (y). It is notedagain that while in the relation y=f(x), x is the independent variableand y is the dependent variable, the reverse is true in the inverserelation x=f (y). As a specific example, if y=arc sin x corresponds toy=f(x) wherein f(x)=arc sin x, then x=sin y corresponds to Prior artautomatic generation of functions by the use of given inverse functionshas been accomplished by automatically solving the equation xf (y)=0using y as the unknown. This system is explained on page 340 of thebook, Electronic Analog Computers, by Gv A. Korn and T. M. Korn,published by McGraw-Hill Book Company, New York, second edition, 1956.The practical success of such equation solving methods is largelydependent on the ease with which f- (y) can be generated. By generationof f- (y) is meant that given a value of y, the corresponding value of f(y) is produced. These methods all give static representation of f- (y),wherein y is time independent.

In contrast to the foregoing automatic equation solving method, themethod of the present invention does not rely on the solving ofequations; and instead of a static representation of f* (y) it uses adynamic representation or time representation of f- (y) by, in effect,replacing y with real time, in which replacement an interval of timerepresents the range of y. To understand this method, an explanation ofcertain terms is appropriate. A time representation of a function =g(x)defined for x x x can be accomplished by letting an interval of timecorrespond to the range of x from x; to x and generating the function=g(x) as a function of time over this interval. Specifically, atransformation of x to the time domain is made by the linear relationwherein t=0 is the instant of time defining the start of the timeinterval referred to above. The size of the time interval is given by Itmust be noted that in each specific instance where the lineartransformation to the time domain is accomplished the variable t islimited in its range from zero to For example if :2x, then a graph ofthe function within its necessarily prescribed limits of x; and x wouldbe a straight line in the ,x coordinate system wherein is the ordinateand x the abscissa and the end points of the straight line would haveordinate Values of 2x and 2x When the linear transformation to the timedomain is accomplished, the abscissa becomes t and the equation becomes=2(kt+x A graph of this latter equation in the ,t coordinate systemyields a straight line whose endpoints again have ordinate values of 2xand 2x The distance between the projections on the t axis of theendpoints is since the abscissa of the lower limit of the function is1:0 and the abscissa of the upper limit of the function is The graphthus terminates very certainly at points determined by the region ofinterest, although t, the independent variable, representing real time,of course continues indefinitely and therefore =2(kt+x could ostensiblybe plotted as a line indefinitely long.

For purposes of this invention a regularly repeated time representationof the function is required. This is called a periodic timerepresentation of the function. In general it is not practical to writean equation for a periodic time representation, although in specificcases it may be simple to do so. The equation above, =2(kt+x representsthe actual equation of only one portion of one cycle of the periodicrepresentation, namely, that portion of one cycle which exhibits thefunctional relationship between a dependent variable and an independentvariable exemplified by the equation =2x wherein x is considered to lieonly between x and x and wherein, correspondingly, varies only from 2xto 2x FIG, 1 shows one example of a periodic time representation of thefunction -2x. This graph would be said to represent the functionalrelation =2(kt+x in the region from t=0 to but it must be observed thatin fact this functional re lation holds only for the segments from a tob, from d to e, from g to h, etc., and then only if t be regarded asstarting at zero at each low terminus, i.e., at a, again at d, again atg, etc. The segments of the graph from b to c, from c to d, from e to 1,from f to g, etc., are not sponding functional value.

.5 sponds to the range of x from x to x From the graph in FIG, 1 it isseen that the basic period of the graph, T, is represented by thelengths ad, dg, etc. As shown, only a portion of each basic period ofthe time representation is occupied by the function, e.g., the timeintervals represented by abscissa lengths ac, df, etc. The functionalrelation is not being represented during a portion of each period shownas the time intervals bcd, efg, etc., each of which has a duration Thepresent invention can use periodic time representations of the typeshown in FIG. 1 wherein the repeated representations of the functionalrelation of interest :are separated by a line on the graph representinga value or values not essentially of interest. The invention can alsouse another type of periodic time representation wherein the functionalrelation of interest effectively occupies the entire period of timeunder consideration. This other type of periodic time representationfalls in two categories: (1) Where the repeated representations of thefunctional relation of interest are contiguous, and (2) where thefunctional relation of interest is contiguous to and alternates with itsmirror image. This latter type of periodic time representation is themost common and the simplest to use and to understand in its behavior inthe practice of the invention. The former type, exemplified in FIG. 1,is sometimes more convenient to produce. An explanation of thegeneration and use of this former type in the invention is set forthhereinafter in relation to the embodiments of FIGS. 26 and 27.

The common term for a device which gives a periodic time representationof a function is a Wave form generator. In contrast to this, the termfunction generator implies a device such that if a value of anindependent variable is introduced, the device produces the corre- Theindependent variable may or may not be varying with time. If the waveform generator produces a periodic time representation wherein eachfunctional display follows its predecessor immediately with no deadinterval between them, the periodic time representation is said to havea 100% duty cycle. In FIG. 1 if the abscissa intervals cd, fg, etc.,were each reduced to zero the representation would have a 100% dutycycle. The actual duty cycle of FIG. 1 is given y 212-21 lcT percent.

The useful terms for describing this invention having been defined, aproof and explanation will now be offered of the novel mathematicalrelation on which are based the method and apparatus of this invention.

A RELATION BETWEEN A FUNCTION AND ITS I vERsE Given a function f(x)defined for af x b, let y denote a particular but arbitrary value of thedependent variable y. Define the variable E as of x. With theappropriate choice of values for the constants A and A depending uponthe nature of f(x),

it develops that for many functions of practical irnpob tance, therelation ave= f- (y0) Evalution of this integral depends, of course,upon the nature of (x), but attention is here restricted to cases wherethe integration is, in the first place, possible. Consider next, then,the values of x in the interval [0, b] which corresponds to y=y that is,the set of x such that f(x) =y Let x and ar denote, respectively, theminimum and maximum values of c in the set f' (y Then Evaluation of thefirst and last integrals presents no problem since the integrands areeither A or A However, the second integral requires carefulconsideration of the nature of the set f' (y Only the case of principalinterest, namely when the set f (y is finite, will :be considered here.Accordingly, if f (y has cardinality n, let x denote the ith value of xin the set Imposing the further restriction that for no x is f(x amaximum or minimum, then, with n even, the first and last integrals of(2) will have the same integrand values, that is, both A or both A2. Ifn is odd, one will be A and the other A We may, therefore, write for Iteven,

Collecting terms and simplifying, the general formula for b I Eda:

can be written as where f(x =y i=1, 2, 11;

1 if ySyo A if y y and 0 e(x a). In the event that y is a maximum orminimum for one or more x then each such x must be treated as two pointswith a corresponding increase in the value of n. Formulas 4a through 4dWlll then apply. It may be noted that results in the several foregoinganalyses would be substantially the same if E were alternatively definedas 1 if y Z/0- 2 i 1/22/0- E 2 if 24 yo K if y=y Where K is a constantof any finite value.

EXAMPLES Here f (y =y and since there is only one real valued root for yn=1. Further, f(x )=f(x e) for every x because x is an increasingfunction. Formula 40 is therefore applicable and leads to the expressionDividing through by (ba) to obtain E Letting A =b and A =a, then E =x x/y Also, f(x -f(x e) 0; hence, Formula 4b is applicable, resulting in(a) For y 0, Relation 4a holds.

Eave=it Al-Az 01xo2 +21rA11 since x =1rx If we choose A =1r/2 and A=1r/2, then (b) For y O, Relation 4b is applicable. Hence,

aw ited A1) out-x02) +2TAZ1 since x =31r-x Again letting A =1r/2 and A=1r/2 as in case (a) the supplement of x (0) For 3 :0, Relation 4c isused. Therefore,

Once again, letting A =1r/ 2, A =-1r/2 and noting that x =O, x =1r, x=21r, the result is =0 as required The significant result in thisexample is that by letting A =1r/ 2 and A =1r/2 for all three casescorresponding to 3/ 0 and y 0, the value of E in each case correspondsto a correct, but numerically smallest member of the set are sin y Onecan therefore Write: E =arc sin y '71'/2 E TF/2. This result isextensively employed in the section on illustrative applications.

Monotonic functions with single valued inverses are readily handled byFormula 4c if the function is increasing (Example 1) and by 4d if it isdecreasing. If the function f(x), a x b, is an increasing one, itfollows from 40 that choice of A =b and A =a makes E =f (y On the otherhand, if f(x) is decreasing, Formula 4d indicates choosing A =a1 and A=b to make E =f (y None of the Formulas 4a through 4d are applicablehowever, to monotonic functions with multiple valued inverses becausethey were developed by assuming all sets f (y) to be finite. Thiscondition of finiteness does not hold for monotonic functions withmultiple valued inverses. Appeal must therefore be made to Equation 2.The integrand of the second integral on the right will be the same asthat of the first, leading to b xoM b j Edx=f A dx+f A da; for a.non-decreasing a a xuM function Monotonic functions and to b XOM b JEdx=f A tlan-Ff A da: for a non-decreasing a a XoM function Hence,

f 1 0M 2(1 .%M fix) non-decreasing a zmM -a) +A (b x f(a:)non-increasing If for f(x) non-decreasing the choice A =b, A =a is made,and for f(x) non-increasing A =a, A =b, then in either case E =x=maximum member of the set J (y0)- In the foregoing examples it wasshown that the appr-opriate choice of values for A and A in the variableE(y, y leads to an average of this variable which is equal to the leastof the inverse values f (y The invention uses this mathematicalprinciple for the following method of generation of a function of asingle variable f(x). By generation of a function of a variable is meantthe production of a physical quantity such as voltage, current,electrical resistance, mechanical displacement or the like whosemagnitude varies in accordance with the variation of the function of thevariable.

METHOD OF FUNCTION GENERATION Object: To generate y=f(x).

Step 1.-Generate a periodic time representation of the inverse functionf" (y).

Step 2.Compare the amplitude of this time function with a given value xof the independent variable of the required function f(x).

Step 3.--Generate, as a result 'of Step 2, a discontinuous functionWhere the values of A and A are time independent, or at least do notvary appreciably over a single period of 0)- Step 4.Take the timeaverage of E(x, x This time average, for appropriately chosen values ofA and A is proportional to the value of the dependent variable ycorresponding to x=x It should be noted that the value of x, namely x ispermitted to change only at a rate which is much smaller than 1/ T, therepetition rate of the periodic time representation of the inversefunction f (y). Also, it should be noted that if the function f (y)happens to be inherently periodic, its period need not correspondidentically with the period, T, chosen for the periodic timerepresentation of f (y) in the method of this invention. For example iff- (y)==sin 0, its period would normally be regarded as 211' radians,constituting the length of the shortest equal sub-interval into whichthe range of the independent variable, 0, can be divided and obtainexactly the same graph of the function in each sub-interval. However, inpracticing the method of the invention, wherein it is required topresent a periodic time representation of f- (y)=sin 0, which involvesthe substitution of (kt-H9 for 0, it is possible within the scope of theinvention to choose a period T for the function sin (kt-Hi correspondingto a range of 6 over only 1r radians. In such a case the periodic timerepresentation of f- (y) would preferably be made up of a repetitivepresentation in regular sequence of only that generally S-shaped portionof the ordinary sine graph lying between 1r/2 and +1r/2.

GENERALIZATION OF BASIC METHOD OF FUNCTION GENERATION The symbolicexpression in Step 3 of the aforementioned method implies at first blushthat it is required to generate (l) E=A during the time interval, say A,throughout which x x and (2) E=A during the time interval, say 6,throughout which x x However, since Step 4 requires taking a timeave-rage of E, it should be apparent to those skilled in the art thatexactly the same end result will be obtained if (1) E is caused to havethe value A not during the time interval, A, wherein x x but during adifferent time interval, say A, so long as A=A'; that is, so long as thelength of time in the interval A equals that in the interval A; and

(2) E is caused to have the value A not during the interval, 6, whereinx x but during a different time interval, say 5', so long as 65:8.

Referring the explanation for simplicity to the occasion of a singletime representation of the inverse functional relation, the significantfact is that the two values A and A of E divide between them an intervalof time equal to the total length of time during which the timerepresentation of the inverse functional relation occurs. Actually, thegeneration of E need not even be simultaneous with the timerepresentation of the inverse functional relation although in practiceit is. The share of time interval assigned to A is equal to the lengthof time that x x and the remainder of the time interval is assigned to AHowever, it is totally immaterial to the value of the end result, namelythe time average of E, whether A takes its share from the first portionof the time interval or from the last portion of the time interval orfrom the middle portion of the time interval or partly fromtwo or moresuch portions.

From the foregoing it is clear that the following is a GENERALIZEDSTATEMENT OF THE METHOD OF FUNCTION GENERATION A during an interval oftime equal to that when mg x A during an interval of time equal to thatwhen x 0 where the values of A and A are time independent, or at leastdo not vary appreciably over a single period of Step 4.Take the timeaverage of E(x, x This time average, for approximately chosen values ofA and A is proportional to the value of the dependent variable ycorresponding to x=x It should be noted that the more extensivelyVerbalized expression E in Step 3 immediately above is fully equivalentto and interchangeable with the more succinct, predominantly symbolicexpression in Step 3 of the earlier recitation of the method. Althoughthe predominantly symbolic expression, being more convenient to write,will be generally used hereinafter, it must be understood andinterpreted always to include the generalized expression.

FIG. 2 shows diagrammatically one apparatus of the present invention forcarrying out the aforedescribed method of function generation.

Numeral 2 indicates a generator of the periodic time representation ofx=f- (y). The output of this generator, being, for example, a voltage orthe like, represented by the expression x(t+T), is fed into an amplitudecomparator 4 into which is also fed the physical quantity such asvoltage, representing x the given value of x for which it is desired toproduce the corresponding value of the function of x. The amplitudecomparator 4 compares the value of x with the value of x generated bythe generator 2 as that value of x varies within the region of interestduring the time cycle. During the period of time when x, the output ofgenerator 2, is less than or equal to x the amplitude comparator 4 putsout a first signal and during the time while the value of x fed into.the amplitude comparator exceeds the value x the amplitude comparatorputs out a second signal. The auxiliary function generator 6 generatesthe discontinuous function E, which function has two values, one valuebeing produced by the generator 6 when the generator 6 is receiving theforementioned first signal from the amplitude comparator 4, and theother, when the generator 6 is receiving from the amplitude comparator 4the aforementioned second signal. The output of the generator 6, whichagain may be an electrical quantity such as a voltage, is averaged by anaveraging device indicted by the numeral 8. When voltages or currentsare involved, such an averaging device can be constituted by a filter.The output of the averaging device 8 is simply the average value of theauxiliary function E and represents, when the proper magnitudes havebeen chosen for the two discrete values of E, the value y of thefunction of x corresponding to the value x of the independent variable.

Generation of x= /2 To illustrate the use of the method of thisinvention, let it be desired to generate the simple function x= /2 whereand correspondingly x x x The inverse function is =2x. Applying themethod of the invention, a periodic time representation of =2x isgenerated. Once such periodic time representation is shown in FIG. 3which happens to have effectively a 100% duty cycle. The amplitude ofthis time function is compared with a given value of the independentvariable of the required function x= /2. Thereupon there is generated,as a result of the comparison, a discontinuous function In this exampleA is assigned the value x and A is assigned the value x The auxiliaryvariable E over one cycle has the value x during the time interval OPand has the value x during the time interval PQ. The time average of Eis then taken over the cycle and this time average will be the value xof the dependent variable x in the original functional relationcorresponding to In FIG. 3 the scale chosen at random happens to havethe following values: x /2=l; x /2=4; 0P=2; PQ=6-. Thus, A :4; A =l; andthe time average over one cycle is given by:

That is, x =1%. To check the validity of the method, a measurement of ismade and it is shown to be 3 /2, which fulfills the equation Thegeneralized concept of the basic method of the invention applied to thegeneration of x= /2 can be seen from the following. In FIG. 3, let therebe established on the t axis a point P located between P and Q, suchthat 0P=PQ. Then, let the generation of the auxiliary variable takeplace in such a manner that E assumes the value A =x during the intervalPQ and assumes the value A =x during the interval OP. Since, under thisconcept, the two values A and A of the auxiliary variable E have dividedbetween them the total time interval 0Q of the cycle of the timerepresentation of the inverse functional relation in the same proportionthat they did in the former case, when A =x occupied the interval OP andA =x occupied the interval PQ, then it is apparent that the average of Eover the full cycle will be exactly the same as in the former case, andwill equal x In this instance, E has the value A not during the intervalof time, OP, when but during the interval of time PQ=0P. Similarly, Ehas the value A not during the interval of time PQ when but during theinterval 0P=PQ. In actual practice with electronic equipment, it isoften more convenient to use an arrangement exemplified by this lattercase, wherein A is generated during the interval P'Q. This isparticularly true when the time representation of the inverse functionalrelation is symmetrical about its intercept on the abscissa axis such asthe sine time function shown in FIG. 4. In such a case, the sum of thetime representation of the inverse plus the given value of theindependent variable changes sign at the point corresponding to P andthis change of sign is useful to control the auxiliary functiongenerator.

It is apparent that, in principle, the method of this invention can bepracticed by generating only a single cycle of the time representation=2(kt+x This will produce a precisely correct value x of the function x=/2q5 so long as 5 remains fixed during the single cycle.

If remains fixed over a plurality of cycles of the time representation,the average of E over all these cycles will still be precisely x If E isaveraged over many cycles, say some thousands of cycles, it will remainindetectibly diiferent from x even though the comparison of with the 5of the time representation be caused to cease at some instant prior tothe exact completion of the last full cycle of the time representation.Since, in practice, it is commonly required to generate values of adependent variable corresponding to numerous values of an independentvariable it is, in practice, desirable to produce a periodic timerepresentation of, e.-g., :2(kt[x so that there will always be at hand acontemporary cycle of this time function against which to compare anexisting value of so as to generate promptly the auxiliary variable Eand hence the ultimately desired value x That is, the most usual case isthe one where takes on various values as time progresses and does notremain fixed at one value.

If changes discontinuously to a new discrete value, say 5 thecorresponding value x could be generated by merely generating oneadditional cycle of the time representation =2(kt+x and performing thecomparison and generation of E as in the first case. However, as justpreviously indicated, it would usually be desirable in conventionalcomputers to produce a periodic time representation, i.e., a continuousrepetition of the cycle, inasmuch as b usually will change with timeand, moreover, will usually change continuously with time. So long asthe value of remains substantially fixed during one cycle of the timerepresentation =2(kt+x the generated function will be substantially xStated in other words, 5 must for accuracy change at a much slower ratethan the repetition rate of the periodic time representation. It, forexample, were itself subject to a periodic variation, then, foraccuracy, the frequency of the variation of should be much less thanthat of the periodic time representation =2(kt+x In practice, if 1/ T isthe repetition rate of the periodic time representation, V this rate or1/ 1001 is usually the maximum rate at which will be allowed to changeto achieve practical computing accuracy. The slower the change in 41 themore accurate will be the corresponding value of x that is produced.

The inverse function is x=sin 0. The sine is an inherently cyclicfunction with limiting values of +1 and 1. A convenient range forconsideration of the function 0=arc sin x is for 1r/2 0 1r/2 since thiscorresponds to the range 1 x1 yielding a sample extending over theentire possible range of the sine. The elementary obvious segment of asine curve to be used for exhibiting a periodic time representation ofthe inverse function x=sin 0 would be the region where 0 ranges from1r/2 to +7r/2 and the equation of one cycle of such a representationwould be x=sin (kt-H2 where 0 =1r/2 and 0 =1r/2 so that t varies from1:0 to

Generation of 0=arc sin x The period of such a cycle is vr/k-O=m-7 k.FIG. 4 shows a periodic time representation of this sine function usingthe elementary segment from 1r2 to '+1r/ 2 as the basic constituent. Thegeneration of =arc sin x for any given value x of x is accomplished inaccordance with the teaching of the invention viz. by comparing thissegmentary time representation over a cycle with x and generating theauxiliary function and then averaging E over the cycle. As mentioned inthe preceding illustrative application, the comparison and averaging canjust as well take place over a plurality of cycles of the timerepresentation and will give the same accurate result. Also, if xchanges with time, the only practical application of the invention is bythe use of a repetition of the cycle of the time representation and thisrepetition must for accuracy be at a rate much faster than the rate ofchange of x The generation of the wave form illustrated in FIG. 4,constituting a repetition of the segment of a conventional sine wavelying between 1r/2 and +1r/2, is certainly possible and can beaccomplished by methods well known in the art as explained, for example,in the volume entitled, Waveforms, No. 19 of the Massachusetts Instituteof Technology Radiation Laboratory Series pub lished in 1949 byMcGraw-Hill Book Co., New York. However, it is readily apparent thateach full cycle of such a- Wave form constitutes one symmetrical half ofthe conventional full sine wave cycle lying between -1r/2 and 31r/2. Itis further apparent that, because of the symmetry, the average value ofE obtained by comparison of x with that half of the conventional sineWave lying between 1r/2 and 31r/2' would be identical with that obtainedby comparison ofx with the segment of a sine wave lying between 7r/2 and1r/2.

Therefore it is clear the same identical accurate result obtained by theuse of the wave form of FIG. 4 can be achieved by using a full sine waveform. The full sine wave form is easily generated by means of a sinewave oscillator and would nomally be less expensive to use than the waveform of FIG. 4.

The use of the entire full wave output of an ordinary sine waveoscillator to generate 0=.arc sin x is now de-v As previously noted, theinverse functionis scribed. x=sin 0. Using conventional symbols, aperiodic time representation of the inverse function employing the fullwave is obtained by setting 0=wt, where t=time and w=angular frequency.The function x=sin wt is, as noted, easily generated by means. of asine, wave. oscillator.

Next, the output of the sine wave oscillator is compared I with a givenvalue of x, denoted by x ;.and as a result of this comparison, "there isgenerated the auxiliary function 1r/2 if sin wtgx r E (Sm l er 2 if sinwt x Because of the previously mentioned syinmetryof .a sine wave, theaverage of E over one cycle of sin wt will be the same as the average ofE over that portion of the cycle lyingbetween 0=1r/2 and- /2. andfurthermore the average of E over one cycle will be the same whether thecycle starts at x=1 or x==Q or elsewhere. Moreover, if the average of Eis taken while x remains substantially unchanged during many cycles ofsin wt, the value of E, will besubstantially the same even' thoughthecomparison of x with x is terminated be-- '14 as before, there are threecases to consider: x 0, x 0 and x 0. For x 0 we have (see FIG. 5);

where 1 t arc sln x mi arc S111 is obtained.

The physical schemes for carrying out the generation of 0=arc sin x, asin all the following examples, are very numerous depending on the natureof the variables and the speed and accuracy requirements. One suchscheme where the variables are voltages, as in electronic analoguecomputers, is shown schematically in FIG. 5. A voltage,

representing sin wt, supplied by any convenient sine wave generator, isapplied to the terminal 10 of an amplitude comparator 12. The amplitudecomparator 12 can be of any convenient form known in the art. Amplitudecomparison and various types of comparators are described in theaforementioned volume Waveforms, especially in chapter 3 and chapter 9.A voltage representing x is supplied to terminal 14 of the comparator.The output of the comparator 12 has two values: one if the comparatorhas found that x x and the other if x x The output of comparator 12 isfed to the generator 16 of the auxiliary function E. The output ofcomparator 12 causes auxiliary function generator E to select one or theother of its two input voltages representing 1r/2 and 1r/2. It selectsthe former if xx and the latter if x x The output E of generator 16 isthen a discontinuous function having the two values constituted by thevoltages representing ar/Z and -n'/2. To average E this output is fedthrough a low pass filter, with cutoff below, the frequency w, whicheffectively takes a time average of E so that the output at terminal 20of the filter 18 is E ave which, as previously demonstrated, equalsarcsin x A compact electrical arrangement of the embodiment of FIG. 5can be made by joining together in one unit the comparator 12 and theauxiliary function generator 16 wherein a polarized or differentialrelay is used, operated by the combination of the voltage at 10 and thevoltage at 14 to make contact alternatively with a source of 1r/2voltage or a source of 1r/2 voltage. Mechanical comparators embodyingthe invention include any of the various forms of differential distanceor angle detectors such as differential gears. Electronic comparatorsand switching circuits would preferably be used when the invention isused in a high speed computer.

Although for simplicity of explanation the input to terminal 10 ofcomparator 12 was shown as sin wt, nevertheless in practice,particularly in conventional electronic computers, it is customary touse voltages of say volts to represent the limiting values of the rangeof a variable. Thus, more generally, the input at terminal 10 would beshown as say x=A sin wt where A might be 100 and A sin wt would be theactual instantaneous voltage at 10. In such a case A x A. Similarly, theinputs at terminals 22 and 24-would more generally be designated 15 ask1r/ 2 and k1r/2. The actual voltage from filter 18 would then be I E =kare sin However, multiplying factors are as readily removed as insertedby conventional procedures and the actual value of the function can thusalways the extracted.

Since arc cos x=arc sin x-1r/2, it sufiices to add 1r/2 to the arc sinefunction in order to obtain the --arc cosine function. This can be donein the circuit of FIG. 5 by adding -1r/2 to the output of generator 16or to the output of filter 18. If the are cosine function is desired itcan readily be produced in the conventional manner known in the art byfeeding arc cosine into an operational amplifier, the output of whichwill then be are cosine. The range is 1r=60.

Of course =arc cos x can also be generated by the use of the method ofthe invention directly without recourse to a modification of the arcsine generator. This could be done by an apparatus similar to that ofFIG. wherein the inputs to comparator 12 would be cos wt and x and theinputs to generator 16 would be 1r and 0 instead of 1r/2 and 1r/2. Itshould be noted that cos wt is, of course, identical in form to sin wtand therefore is obtained from an ordinary sine wave oscillator, whichcan, as well, be called a cosine wave oscillator. The function thengenerated by generator 16 would be {0 if x g 00 71' if x x This is forthe range 060611:

The can be done for sin 0 in one of two ways: (A) by placing the arcsine circuit of FIG. 5 in the feedback of an amplifier; or (B) by adirect application of the method of the invention. Both methods areeasily adapted to the generation of cos 0. Method A is illustrated inFIG. 6 and Method B is shown in FIG. 7.

In FIG. 6 numeral designates an arc sine generator identical to theentire assembly of FIG. 5 which receives sin wt at one input terminal 26and receives y at its other input terminal 27 and yields arc sin y atits output terminal 28. The output of the arc sine generator, and avoltage representing 0, applied at input terminal 29, are each fedthrough separate identical resistors R to the summing junction 30 of anoperational amplifier 32. The output of this amplifier at 34 will be aquantity such that its arc sine equals +0. This quantity is then sin 0.This arrangement is operative in the region from 1r/2 to 11'/ 2. Bythrowing the switch 36 from the zero position to the position Where 1r/2is fed into summing junction 30 through another resistor R, of the samevalue as each of the aforementioned two resistors, the output of theapparatus becomes sin (01r/2) which equals cos 0. If cos 0 is desired,it is a simple matter to feed the output at 34 into an amplifier toreverse its sign. It should be noted that the range of the device ofFIG. 6 when used to generate a cosine function is from 0 0 wr.

In FIG. 7 an apparatus using the direct application of the method ofthis invention is shown. A comparator 38 is supplied at terminal 40 withz(t) a periodic time representation of the arc sine function of thevariety shown in FIG. 8, for example. The voltage representing 6, whosesine or cosine is ultimately to be produced, is fed into terminal 42.The comparator compares the two voltages at terminals 40 and 42 and thenactuates auxiliary function generator 44, which is supplied withvoltages at terminals 46 and 48 representing +1 and 1, so that generator44 generates Generation of 0=arc cos x Generation of sin 0 and cos 6'The output of generator 44 is averaged by running it through a low passfilter 50 whose cutoff is below frequency 1/ T but high enough to havelittle effect on the maximum frequency of change of 0. The output offilter 50 at terminal 52 is then y sin 0 where By throwing switch 54from the zero terminal to the 1r/2 terminal, the independent variableinput to the comparator becomes 0-1r/2 instead of 0 and the device willbe made to produce y=sin (01r/2=COS 0 where 061n As previously mentionedcos 0 can easily be converted into cos 0 by feeding it through anamplifier.

The periodic time representation of the arc sine function can beobtained in a variety of ways for use in Method B. Among these are:

(a) Harmonic synthesis of time sine functions which is'simply thereverse of harmonic or Fourier analysis:

(b) Harmonic modification of a square wave which amounts to filteringout from a square wave (which contains practically all frequencies) suchfrequencies that those which remain produce the desired time function;

(c) Letting the x input in FIG. 5 be a triangular wave form of amplitude+1 and 1 and of repetition rate much less than w. That is, x can bevaried as a triangular function of time and the output of terminal 20 ofsuch a device as FIG. 5 would then be a periodic time representation ofarc sin x ((1) Direct modification of a periodic time function, such assin wt, with a diode function generator.

The last mentioned tiem is shown in FIG. 8 where sin wt is beingmodified to a time function that gives the values of the arc sinebetween 1r/2 and 1r/2 in a periodic manner. FIG. 9 shows the staticfunction that would have to be set up on a diode or similar functiongenerator to so modify sin wt. The use of diode function generators andthe like in this manner to accomplish modification of functions is fullyset forth in Korn and Korn op. cit. page 2.90 if.

As previously noted, in the illustrative sine and cosine generators ofFIGS. 6 and 7, the range of 19 is 1r/2 to 1r/2 for sin 0 and 0 to 1r forcosine 0. These rangm can be extended by appropriate modification of theequipment when 0 exceeds these ranges. One example of an actual circuitexhibiting such a modification is shown in FIG. 10. This circuit can besaid to represent essentially an actual circuit exemplifying theschematic arrangement of FIG. 6 plus the modification employed to extendthe range of 6 to from 31r/2 to 31r/2 for sin 0 and to -1r to 211- forcosine 0. The circuit comprises a comparator including an operationalamplifier 54 having two input terminals 56 and 58 into which are fed,respectively, sin wt and y for comparison. The limiter connected toamplifier 54 is arranged to produce at the output terminal 60 adiscontinuous voltage function having only two values, say +2 if sinwt+y 0, and 2 if sin wt+y 0. This voltage is chosen as being sufficientto cause diode 62 either to conductor or not to conduct. The circuitfurther comprises an auxiliary function generator and an averagingdevice for its output including diodes 62 and 64, operational amplifier66 with input terminals 68 and 70, filter circuit 72 and output terminal74.

If sin wt +y, the plate of diode 62 is made negative and therefore diode64 will conduct and the net voltage appearing at terminal 76 will bethat due to 1r/ 2 from terminal 68 minus, from terminal 70, 1r/ 2increased by virtue of R 2 to 1r so that the net effect at terminal 76will be that of 1r/ 2. When sin wt y, the net voltage at terminal 76will be that due to effectively +1r/ 2. The output at 76 isaveraged bythe filter circuit 72 so that are sin y appears at terminal 74.

To produce the sine of 0, it suffices to embody the aforedescribed arcsine generator in the feedback of an amplifier circuit in the manner ofFIG. 6. In FIG 10 the output 74 is placed in the feedback of operationalamplifier 78, whose output at terminal 80 provides the y to be fed intothe arc sine generator at terminal 58. 0, whose sine it is desired togenerate, has its negative applied at terminal 82 and joins the outputof the arc sine generator at summing junction 84 serving as the inputsource for amplifier 78. Since the entire monotonic section of the sinecurve is represented by the portion lying between =1r/Z and 0=1r/ 2, theaforedescribed circuit will generate accurately the value of sin 0 forany 0 lying with these limits. As thus far described, the constructionand operation of the circuit is substantially identical with that ofFIG. 6. In the circuit of FIG. 6, and its counterpart in FIG. 10, if thevalue of the independent variable input 0 is allowed to exceed thelimits 7r/ 2 and 1r/ 2, then the output of the device, i.e., terminal 34in FIG. 6 or terminal 80 of its counterpart in FIG. 10, would go veryhighly negative or positive until the amp-lifier saturates and thusgives an erroneous reading. The reason for this erroneous reading isthat the maximum voltage which the device, as thus far described, cansupply at terminal 74 in FIG. 10, for example, is -1r/2 or +1r/2 andthis is sufficient to balance at junction 84 only 1r/2 or +1r/2originating at terminal 82. If the dif ferenoe between these twovoltages appearing at 84 is not very close to zero, the tremendousamplification of amplifier 78 causes its output at 80 to rise until theamplifier saturates.

To extend the limits of the function would require some modificationwhich would cause the output at terminal 80, which is, for example say+1 when 0 is 90, to decrease when 0 increases to, say 93, until itreaches the same value that it had when 0 was 87", since In other words,the circuit of FIG. 6 and its counterpart in FIG. 10 can be made toproduce a correct value for the sine of 0 with 0 equal to, say 93, ifthe effective 0 input were made 87 or in general if the effective inputwere reduced to a value 02(01r/2). This is accomplished in FIG. 10 byadding the two additional branches 86 and 88 to be used underappropriate circumstances to contribute to the voltage at terminal 76.

The operation of the circuit can easily be understood by reference toFIG. 11 in which the solid line representation is a graph of effectiveinput to junction 84 in FIG. 10 versus 0, which latter is applied toterminal 82. As the input of 0 at terminal 82 goes'from 1r/2 to 1r/2 theeffective input at junction 84 must go from 1r/2 to 1r/2 and it does so,as illustrated in FIG. 11 by the line segment PQ, by virtue of theoperation of the circuit heretofore described as the counterpart of FIG.6. As 0 increases beyond 1r/2 and the input 9 at terminal 82, designatedas 6 becomes more negative than 1r/2, it is required for the effectiveinput at 84, designated as 0 to decrease in absolute magnitude to thevalue given by the equation 0 =0 +2(9 1r/2). The reason for this can beseen from an example using actual numbers. When say 0 =87, the output atterminal 80 is sin 87. When -0 =90, the output at ter- \minal 80 is sin90. However, if -0 should be allowed to become more negative to say 93,then the system, which is built to work only within the limits -1r/2 to1r/2, cannot handle the -93 voltage and, so to speak, goes berserkyielding an output at 80 representing saturation of amplifier 78. But,observing that sin 93=sin 87, it is apparent that if, when 0 =93, -0 canbe made equal to 87, then the apparatus, which is fully capable ofhandling a voltage of -87 at terminal 84 without going berserk, willyield at terminal 80 a voltage equal to sin 87". This latter, of course,is numerically equal to sin 93 so that the apparatus is, in effect,handling a voltage input at 82 representing 0 1r/ 2.

It should be noted that the general requirement, previously stated, thatfor 0 1r/2, -0 must is represented in the P ng numerical example thu Toaccomplish this requirement means contributing, at the time when 0 1r/2, a component at 84 which will add, to the component at 84 due to 0 theeffect of 2(0 1r/2) applied through an input resistor equal in size to85. This added component arrives from the network comprised of branches86 and 88. The same voltage 0 applied to terminal 82 is alwayssimultaneously applied to terminal 90. When 0 at terminal 90 is morenegative than 1r/ 2, the potential of the cathode of diode 94 is loweredbelow that of its plate and hence diode 94 conducts, causing a currentto flow in branch 86 whose magnitude is proportional to 0(1r/2) dividedby R /2. This, in effect, contributes at junction 76 a potential of2(0+1r/2) which, in passing through amplifier 66, changes its sign and,since resistor 95 equals resistor 85, appears at terminal 84 as,effectively, 2(0-1r/2), compared to the 0 at the same terminalcontributed from terminal 82. The net or effective input, then, atterminal 84 upon initiation of the operation is If, as in theaforementioned example, 0:93", then the net effective input at terminal84 would correspond to 93-11-:87, a magnitude which is within the limitsof -1r/2 to +1r/2 under which the circuit is capable of giving correctresults. The production of the proper efiective input at terminal 84 forthe region 1r/2 a31r/2 is shown graphically in FIG. 11 by the dottedline segment PR, representing the contribution from 0 the dash-dot linesegment ST, representing the contribution from branch 86 equal to 2(01r/2); and the solid line segment PU representing the sum of the twocontributions at terminal 84.

An analogous situation occurs with conduction in branch 88 when 31r/2 0vr/2. This is shown graphically in FIG. 11 by line segments QV and LMwhich add to produce QN.

This circuit can be used, by throwing switch 97 to the 1r/2 position,for generating cos 0 for the limits 1r 6 21r. But, of course, it isoperable only within these limits for the -cos 0 (and 31r/ 20 31r/ 2 forthe sine) for the reason that these limits are necessary, with thiscircuit, to maintain the effective net input at 84 between -1r/2 and7r/2. If 0 should exceed 31r/2, e.g., should be 271, then the netefiective input at 84 would be 01r=271180=91 which is beyond theoperating limits of the circuit. However, further extension beyond therange -31r/2 to 31r/2 for the sine and -1r to 271" for the cosine is, ofcourse, possible using the illus-' trated principle, i.e., by energizingappropriate circuits whenever the absolute magnitude of 0 exceeds 31r/2,51r/2, etc., so that the effective input at 84 is always maintained inthe range 1r/2 to +1r/2.

The circuit of FIG. 7 using Method B can also be modified to extend therange of 0. FIG. 12 illustrates such a modification showing oneparticular embodiment. When operating in the range of 1r/2 0 1r/2, thecircuit compares 0 applied at terminal 102 with the time representationz(t) of the arc sine function applied at terminal 104 and, on the basisof the comparison, selects, in a manner similar to the operation of thecircuit of FIG. 10, either +1 or -1 from terminals 106 or 108 as thevalue of the auxiliary variable. The auxiliary variable is averaged bythe filter 110 yielding sin 0 at output terminal 112. If 0 exceeds 1r/2,diode 114 conducts and produces as the effective input at terminal 116the sum Of 0+2(01r/2)=0-1r, the first term on the left hand side beingdue to branch 118 and the second term being due to branch 120. This isso because resistor 119 is twice as large as resistor 121. So long as031r/2, the quantity 01r effectively applied at 116 remains within the-11-/2 to 1r/2 range of effective inputs within which the circuit givescorrect results. Similarly, when 1r/2 branch 122 conducts and thecircuit yields correct results for (i -31M 2. If switch 124 is swung tothe 11'/ 2 terminal the circuit operates to generate cos 0 for n' fizn'. As indicated in the discussion of FIG. 10, the circuit of FIG. 12can, of course, be extended using the illustrated principle beyond therange -31r/ 2 i9 31r/2 for the sine and w 0 21r for the cosine.

Generation of sin 0, cos 0 with unlimited angular range It is oftenimportant in problems using angles to have an unlimited angular rangewhen generating sine or cosine functions. The circuits of FIGS. 6, 7,10, and 12 can be adapted to this requirement through the use of anauxiliary circuit. This auxiliary circuit makes use of dB/dt to producean oscillation that sweeps through the restricted angular ranges of thesine and cosine generators (e.g., for one sine generator the range wouldbe from 1r/2 to +1r/2) at a rate proportional to de/dz. When dB/dt isconstant this oscillation becomes an isosceles triangular wave. Thecircuit, when used for example to supply a sine generator, will thensupply the sine generator with an input 0 which always lies between 1r/2and +1r/2 and at each instant has a value such that its sine is equal tothe sine of the actual angle 0 (which is the actual machine variable) atthat instant. That is, the circuit in a sense performs a function whichresults in the mathematical equivalent of converting the actual 0, nomatter how large it may "be, into an angle in either the first or fourthquadrants having an equivalent sine. The circuit performs this functionwithout receiving (after initiation of its operation) any actual 6 inputbut by receiving merely actual dH/dt input which latter it integrateswith respect to time in order to be able to sense increments of actual0. A preferred embodiment of the auxiliary circuit is shown in FIG. 13.

The circuit of FIG. 13 comprises an operational amplifier 126 whoseoutput at terminal 128 will ultimately be the desired 0 whose negativewould be fed into, for example, terminal 29 of the sine generator ofFIG. 6 or the like. The amplifier 126 is shunted by a condenser 130. Thecapacitor-shunted amplifier 126, 131) is located in one branch 132 of aparallel circuit including another branch 134, which parallel circuit isconnected in series with a pair of operational amplifiers 136 and 138.Amplifier 136 is shunted by alternatively operating branches 140 and142, the former branch including a diode 144 and a voltage source suchas a battery 146 of value 1r/ 2, and the latter branch including a diode148 and a voltage source such as a battery 150 of such a value as toproduce at terminal 152 a voltage of -11'/ 2 when branch 142 isconducting.

Branch 132 includes two resistors 154 and 156 of equal value at whosejunction 158 is connected the output of a circuit yielding angular rateof change. This angular rate circuit receives at its input terminal 160the quantity dO/dt, the time rate of change of the actual machinevariable 0, which it can apply to junction 158 when diode 162 isconducting. Alternatively, when diode 164 is conducting, the angularrate circuit can apply dQ/a't to junction 158, the negative beingobtained by simply passing dH/dt through the amplifier 166.

To initiate the operation of the circuit of FIG. 13, both 0 and dfi/dtmust be initially available but, after initiation of the operation, allthat is needed is d0/dt and no further need exists for information as tothe value of the actual machine variable 0 to enable the device tocontinue functioning. The operation of the device proceeds as follows.At time t=0, 0, the quantity appearing at terminal 128, is assumed to be0(0). This value is established by applying, either automatically ormanually, a voltage across capacitor 130, this being the initial valueof the actual machine variable 0. This voltage can be applied by simplyplacing a battery of the correct value across the terminals of condenser130, it being remembered thatthe potential at the summing junction 168of the operational amplifier 126 is always substantially zero. orground. At the same instant that the initial value of 0 is appliedacross condenser 130, dH/dt is connected to terminal 160. At time t=0+5,the battery imposing 6(0) across condenser 130 is removed. While thebattery was in position across condenser 130, the potential across thecondenser was necessarily maintained constant. Upon removal of thebattery, however, the amplifier 126 with its associated condenser actsas an integrator and begins to integrate its input voltage which isapplied to one or both of its input resistors 154, 156. Assuming that00(0) 1r/2, it will be intended for the integrator to add to the initialvalue 0(0) appearing at 128 the increment represented by the integral ofd0/ dz over a period of time until the value of 0 at 128 reaches 1r/ 2.To insure that the initial operation is started in the right directionto perform this addition, it is required that, at the start of theoperation, a positive input should exist at input terminal 188 to theamplifier 136. This can easily be accomplished by throwing the switch188 to a source such as 186 of positive potential, which could be forexample merely one volt, at'the instant of the start of the operationand then throwing it back into the solid line position very rapidly,using a makebefore-break switch if desired. The reason for applying aninitial positive potential at 188 can be seen from the followinganalysis.

With, say, at 128 from the starting battery applied across 130, therewould be experienced at summing junction 174 the effect of +80 from 128plus the effect transmitted from terminal 152. At 152 there will,however, be a voltage of 1r/2 produced by virtue of the followingsequence of events. When terminal 188 is connected to the positivebattery source 186, the output of amplifier 136 at 152 will be negative.By virtue of battery 150 and diode 148, it is held at a negative levelof 1r/2. Therefore, at summing junction 174 there will be felt theeffect of, say, +80 from 128 combined with from 152 giving a netnegative effect at 174 which will emanate with a change of sign as apositive voltage at 172. This positive voltage at 172 is fed, throughresistor 189, into surnming junction 170, thus maintaining the circuitin a stable state since the positive starting voltage at 188 from thebattery 186 was precisely the sign required to produce a positivevoltage at 172 to be fed into 188 so that the device will beself-maintaining.

With -1r/Z appearing at junction 152, as just described, the potentialat junction 158 will be 1r/4 since resistors 154 and 156 are equal andthe potential at 168, as previously indicated, is substantially zero.dQ/dt is assumed to have a value between zero and 1r/4. The presence of1r/4 at junction 158 therefore causes diode 164 to conduct, thereuponclamping the voltage at junction 158 at the level of de/dr which mightbe at, say, 40 volts. With 40 volts at terminal 158, the integratingamplifier 126 will integrate this voltage continuously as long as it isapplied at terminal 158, resulting in an increase in the positivevoltage at terminal 178 and hence, at 128. When the voltage at 128 hasreached 1r/2, a change will occur. As soon as the voltage at 128 exceedsever so slightly 1r/ 2, the net effect at junction 174 will flip fromnegative to positive. For example, +91 at junction 128 combined with1r/2 from junction 152 will yield a net effect at 174 of +1. Thispositive voltage at 174 changes its sign by passing through amplifier138, and the voltage at 172 will then be negative. A negative voltage at172 fed into junction 170 will produce a positive voltage at 152, whichpositive voltage will be fixed at 11'/ 2 by the limiting effect ofbranch having battery 146 and diode 144.

As soon as +1r/2 appears at 152 this will tend to produce at junction158 a potential of +1r/4 which instantly stops diode 164 from conductingand causes diode 162 to conduct, transmitting to junction 158 thevoltage dO/dt

2. AN APPARATUS FOR GENERATING AN EXPONENTIAL FUNCTION OF THE FORM BEKXOF AN INDEPENDENT VARIABLE X WHICH COMPRISES AMPLIFIER MEANS; MEANS FORSUPPLYING TO SAID AMPLIFIER MEANS THE NEGATIVE OF SAID INDEPENDENTVARIABLE AS PART OF THE INPUT TO SAID AMPLIFIER MEANS; MEANS FORSUPPLYING TO SAID AMPLIFIER MEANS A SECOND PART OF THE INPUT TO SAIDAMPLIFIER MEANS COMPRISING A CIRCUIT FOR PRODUCING A PERIODIC TIMEREPRESENTATION OF AN EXPONENTIAL FUNCTION OF THE FORM BEKT WHEREIN K ISA CONSTANT AND T IS TIME, A COMPARATOR, MEANS FOR SUPPLYING SAIDEXPONENTIAL FUNCTION OUTPUT OF SAID CIRCUIT TO SAID COMPARATOR, ANAUXILIARY FUNCTION GENERATOR CONTROLLED BY THE OUTPUT OF SAID COMPARATORFOR GENERATING AN AUXILIARY VARIABLE HAVING ONLY THE TWO VALUES ZERO AND